Saturday, December 11, 2010

Shermer gives 8 candidate explanations in “The Biggest Big Question of All” [Online]. The problem with these candidate explanations is they each assume the laws of physics. But the question is: why were those laws selected for actuality? Shermer’s candidate explanations do not provide an answer. Here is a 2-page summary of another attempt at an explanation for existence. However, it, too, fails.

There are regularities in the behavior of matter. But what is matter? It is consistent with the regularities that matter is ultimately mathematical structure. We take this as a working hypothesis.

Hypothesis: the universe is ultimately mathematical structure.

Note this hypothesis is consistent with materialism but inconsistent with dualism.

The logical place to start is with the question: why is there something rather than nothing? The counter-move is to note this question just assumes that nothing is more natural (or makes fewer assumptions) than the existence of something. []. But is that assumption justified? Might it not turn out to be the case that the existence of something is more “natural” than nothing?

Because of the hypothesis, it is sufficient to consider the case where the universe is definable as some formal mathematical system. Call this system T. The question is: why does T exist?

Many formal systems can express things about themselves. Suppose T is a formal system that permits self-reference. Suppose it is equipped with the existential quantifier and the necessity modal operator □. Then we want (for reasons given below) the minimal system that entails that T necessarily exists. This will be T itself. So we have the first condition on T:

(1) TT

T entails that T necessarily exists.[1] The point is this. If the actual physical universe were T, it would be logically necessary that T exists. T does not merely represent the universe, it is the universe. If T satisfies (1) it would be a physical fact that the universe is logically necessary. If we knew our universe to be T, we would have an explanation for existence.

But why would T exist in the first place? Perhaps existence is inherently relational. The number 2 exists relative to the number 4, and this chair exists relative to this table.[2] Assuming this notion could be worked out, it would be sufficient that T talk about itself, since it is definitionally related to itself.

Ideally is there a unique maximal T. [Tenneson] . If so, one could pursue the thesis that T’s being related to its own necessary existence would make it the “most likely” structure to exist, out of all possible mathematical structures. In principle one could compare the structure T with the observed physical laws of the universe and get confirmation or refutation the universe is T.

The problem is this. Consider the two sentences:

(2) This sentence exists.

and

(3) This sentence does not exist.

Evidently, the two sentences (2) and (3) exist equally. But (2) is true to the extent (3) is false. So the content of a sentence does not have anything to do with its existence.

The upshot is that the program of justifying the existence of T, as being the unique theory that “bootstraps” itself into existence, fails.[3] Even in a Platonic sense T does not exist any more than a theory which expresses it does not exist.

Starting from the example above, it is interesting to ask what is the most general setting for which one could give a “proof” there is no structure-related explanation for existence.

[1] In fact T should imply that the existence of T is logically inevitable, which is an even stronger statement then that T necessarily exists.

[2] Does the collection of things that exist form an equivalence class?

[3] There might be a loophole in the theorem that if T permits self-reference and has strong negation then it cannot contain a truth predicate, but I don’t know. [Tarski]

Tuesday, December 7, 2010

The problem with the solutions 3-10 for existence in The Biggest Big Question of All is they assume the laws of physics. But why were those laws selected for actuality? They do not solve the mystery.

Here is a 1-page summary of a more sustained attempt at an answer. However, it, too fails.

There are regularities in the behavior of matter. But what is matter? It is consistent with the regularities that matter is ultimately mathematical structure. We take this as a working hypothesis.

Hypothesis: the universe is ultimately mathematical structure.

Note this hypothesis is consistent with materialism but inconsistent with dualism.

The logical place to start is with the question: why is there something rather than nothing? The counter-move is to note this question just assumes that nothing is more natural (or makes fewer assumptions) than the existence of something. [Stenger]. But is that assumption justified? Might it not turn out to be the case that the existence of something is more “natural” than nothing?

Because of the hypothesis, it is sufficient to consider the case where the universe is definable by some formal mathematical system. Call this system T. The question is: why does T exist?

Many formal systems can talk about themselves in some sense. Suppose T is a formal system that permits self-reference. Suppose it is equipped with the existential quantifier and the modal operator “necessity”: □. Then we want the minimal system that can assert that T necessarily exists. This will be T itself. So we have the first condition on T:

(1)

T expresses that T necessarily exists. Then, if the actual physical universe were T, it would be logically necessary that T exists. T does not merely represent the universe, it is the universe. If T satisfies (1) it would be a physical fact that the universe is logically necessary. If we knew our universe to be T, we would have an explanation for existence.

But why would T exist in the first place? Perhaps, the argument goes, existence is inherently relational. It is then sufficient that T talk about itself, since it is definitionally related to itself.

Ideally is there a unique maximal T. [] . If so, we pursue the thesis that T’s being related to its own necessary existence would make it the “most likely” structure to exist, out of all possible mathematical structures.

The problem is this. Consider the two sentences:

(2) This sentence exists.

and

(3) This sentence does not exist.

Evidently, the two sentences (2) and (3) exist equally. But (2) is true to the extent (3) is false. So the content of a sentence does not have anything to do with its existence.

The upshot is that the program of justifying the existence of T, as being the unique theory that “bootstraps” itself into relational existence, fails.[1] Even in a Platonic sense T does not exist any more than a theory which expresses it does not exist.

It is interesting to ask if there is a “proof” there is no structure-related explanation for existence.

[1] There might be a loophole in the theorem that if T permits self-reference and has strong negation then it cannot contain a truth predicate, but I don’t know. []

Monday, October 18, 2010

Out of whatever it is that could exist, what is the relationship between content (what it is that would exist) and the likelihood of it existing? Two motivations for considerating this question are its intrinsic interest and how it casts the possibility of an explanation for existence.

If something is inconsistent the probability of its existing is 0. (Bill Vallichela)

What is the likelihood of something that is consistent existing?

I mention four cases 1. the actual universe, 2. possibilities 3. the sentences of a mathematical structure 4. quantum possibilia

1. the actual universe

We want an explanation not only for the existence of something, but for the specific universe we inhabit. In this case one would have to argue that our universe is that unique entity that could not not exist.

It is logically inconsistent that the universe not exist. But that would seem to be contingent and not a priori true.

The dualist would say I am giving too much weight to logic.

2. possibilities

What is the likelihood of the existence of a pink elephant outside my door right now? Given sufficient other information about the actual universe, there is a 0 probability. But why couldn’t there be a universe in which there things are pretty much the same but there is a pink elephant standing outside “my door”. (I put “my door” in quotes because I don’t know if it refers to the same ontological entity or not: there are two doors, one in each universe).

It has been speculated that all consistent things exist (see e.g. Tegmark()). This would at least give the answer to our question that consistency is necessary and sufficient for existence. This may or may not be true, but it leaves open the question of why something should exist just because it is consistent.

3. sentences

What if mathematical structure ontologically exists? Such structure is codified by the sentences of a formal system.

Might a sentence exist relative to itself? For the self-referential case we have

(1)This sentence exists.

(2)This sentence does not exist.

To the extent (1) is true (2) is false, but they exist equally. So the truth of a sentence does not seem to affect its existence. Therefore, it would be hard to argue that the content of a sentence makes any difference to its existence. As a result they all have equal likelihood of existing.

This would seem to rule out one kind of explanation for existence. The explanation involves the following. The universe is ultimately just mathematical structure. In fact, the universe just is the maximal theory T which implies that it necessarily exists. (Then, if we knew our universe to be such theory T, the necessary existence of our universe would a physical fact, and we would have an explanation for existence.) Without going in to it further, the problem is already apparent. Existential statements in T are irrelevant to the existence of T, as witnessed by (1) and (2).

What if sentences are not self-referential? I don’t know.

4. Conclusion

It doesn’t look good for an explanation for existence. On the other hand, I don’t know of a proof there is no explanation for existence (for some reasonable general class of notions of proof).

Friday, October 15, 2010

Is so relative to itself in the context of its (necessarily) possible definition...

But isn’t the definition contingent?

What we require is that in every possible world it is possible that (1) exists. But it is probably better not to use possible-worlds semantics at all. Then, (it would have to be argued) this possibility might have an ontological status indistinguishable from actual “self-apparent” existence. […]

So why wouldn’t every self-referential sentence exist? Compare (1) to

(2)This sentence does not exist.

To the extent (1) is true, (2) is false, but they exist equally. So the truth of the sentence does not seem to affect its existence. Therefore, it would be hard to argue that content makes any difference to a sentence’s existence.

Thursday, October 14, 2010

Ifso, we could have a hierarchy of theories based on how likely they are to exist. Such a hierarchy might be based on something like the following. Starting with the least likely to exist, consider T such that

(1) T ¢Ø$T

I don’t know, hopefully there is no such T. If there were (and it was consistent) it would render the notion of existence within a theory irrelevant to the notion of actual existence. If there is no such T the notion of existence from within a theory is possible.

(2) T¢($x)(x=x)

This T is the default degree of existence, because so many theories have this property. I don’t know if this is still valid when T is self-referential:

(3) T¢$T

This T would be still more likely to exist.

(4) T¢ $T

This T would arguably be the most likely of all these theories to “exist”.

--Are there conditions stronger than (4)?

We want to rule out theories for which (2), but rule in theories for with (4). This is because we want to rule out the idea that all consistent mathematical structures exist. This would not be an explanation as to why any of them exist. Given that they all have the same existential status, why is it not the case that none of them exist?

One clue as to a way to proceed is that we want the why of its existence to be inextricably involved with its form (content). What does T have in (4) that it does not have in (2)? One thing is self-reference, but many sentences are self-referential. The salient property is condition (4), namely T stands in relation to the logical inevitability of its existence. Why should that matter? Because T’s existence is a necessary possibility.

(I have to remind myself: the reason we want condition (4) in the first place is that if we knew our universe was theory T, then it’s existence would be logically inevitable. The necessary existence of the universe would be a physical fact.)

What is a necessary possibility? Well, what is a possibility? Well, what does it mean to say such-and-such could be the case? In what ways does a possibility exist?

It seems plausible to say that a possibility exists the same way a number exists. Both of these are descriptions of reality.

Anyway, the first thing we do away with is possible-worlds semantics. Suppose we say í$T. It is not true that $T in some worlds and Ø$T in some disjoint set of worlds. What is true “in all possible worlds” is í$T. (Recall it is possible because we make no assumptions, so we do not assume nothingness is more natural than the existence of something.) Apparently

(5)í$T® í$T

I don’t know if this respects the true-in-a-world and true-of-a-world distinction.

Then, the argument proceeds, the existence of the possibility of T is sufficient, in view of its peculiar definition, for it to bootstrap itself in to a valid notion of existence.